Theorem sltssn 27840

Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)

Hypotheses

Ref	Expression
sltssn.1	⊢ (𝜑 → 𝐴 ∈ No )
sltssn.2	⊢ (𝜑 → 𝐵 ∈ No )
sltssn.3	⊢ (𝜑 → 𝐴 <s 𝐵)

Assertion

Ref	Expression
sltssn	⊢ (𝜑 → {𝐴} <<s {𝐵})

Proof of Theorem sltssn

Step	Hyp	Ref	Expression
1		sltssn.3	. 2 ⊢ (𝜑 → 𝐴 <s 𝐵)
2		sltssn.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		sltssn.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
4	2, 3	sltssnb 27839	. 2 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
5	1, 4	mpbird 259	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ∈ wcel 2141  {csn 4581   class class class wbr 5099   No csur 27681   <s clts 27682   <<s cslts 27827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-slts 27828

This theorem is referenced by:  cutneg  27886  zcuts  28477  twocut  28493  nohalf  28494  pw2recs  28508  halfcut  28528  addhalfcut  28529  pw2cut2  28532  bdaypw2n0bndlem  28533  bdayfinbndlem1  28537